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Global dynamics of a Holling Type-III two prey–one predator discrete model with optimal harvest strategy

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Abstract

This paper deals with a discrete-time two prey–one predator system with Holling Type-III functional response, along with inter-specific competition between the prey and intra-specific competition among the predators, with nonlinear harvesting of a prey population. The conditions for existence and local stability of the equilibrium points and the sufficient conditions for global asymptotic stability of the interior equilibrium point are derived. Furthermore, optimal harvesting policy is obtained by extension of Pontryagin’s maximum principle to discrete system. Meanwhile, some numerical simulations are provided to corroborate the analytical results.

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References

  1. Danca, M., Codreanu, S., Bako, B.: Detailed analysis of a nonlinear prey–predator model. J. Biol. Phys. 23, 11–20 (1997). https://doi.org/10.1023/A:1004918920121

    Article  Google Scholar 

  2. Xu, C., Shao, Y.: Bifurcations in a predator–prey model with discrete and distributed time delay. Nonlinear Dyn. 67, 2207–2223 (2012). https://doi.org/10.1007/s11071-011-0140-1

    Article  MathSciNet  MATH  Google Scholar 

  3. Murdoch, W.W.: Switching in general predators: experiments on predator specificity and stability of prey populations. Ecol. Monogr. 39, 335–354 (1969)

    Article  Google Scholar 

  4. van Baalen, M., Křivan, V., van Rijn, P.C.J., Sabelis, M.W.: Alternative food, switching predators, and the persistence of predator–prey systems. Am. Nat. 157, 512–524 (2001). https://doi.org/10.1086/319933

    Article  Google Scholar 

  5. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 97, 5–60 (1965). https://doi.org/10.4039/entm9745fv

    Article  Google Scholar 

  6. Zhuo, X.L., Feng, F.X.: Stability for a new discrete ratio-dependent predator–prey system. Qual. Theory Dyn. Syst. 17, 189–202 (2018). https://doi.org/10.1007/s12346-017-0228-1

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhuo, X.: Global asymptotic stability for a two-species discrete ratio-dependent predator–prey system. Int. J. Biomath. (2013). https://doi.org/10.1142/S1793524512500647

    Article  MathSciNet  MATH  Google Scholar 

  8. Mishra, P., Raw, S.N.: Dynamical complexities in a predator–prey system involving teams of two prey and one predator. J. Appl. Math. Comput. (2019). https://doi.org/10.1007/s12190-018-01236-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Hu, Z., Teng, Z., Zhang, T., Zhou, Q., Chen, X.: Globally asymptotically stable analysis in a discrete time eco-epidemiological system. Chaos Solitons Fractals 99, 20–31 (2017). https://doi.org/10.1016/j.chaos.2017.03.042

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, G., Teng, Z.: On the stability in a discrete two-species competition system. J. Appl. Math. Comput. 38, 25–39 (2012). https://doi.org/10.1007/s12190-010-0460-1

    Article  MathSciNet  MATH  Google Scholar 

  11. Kar, T.K., Matsuda, H.: Global dynamics and controllability of a harvested prey–predator system with Holling type III functional response. Nonlinear Anal. Hybrid Syst. 1, 59–67 (2007). https://doi.org/10.1016/j.nahs.2006.03.002

    Article  MathSciNet  MATH  Google Scholar 

  12. Sengupta, S., Das, P., Mukjerjee, D.: Stochastic non-autonomous Holling type-III prey–predator model with predator’s intra-specific competition. Discrete Contin. Dyn. Syst. B 23, 3275–3296 (2018). https://doi.org/10.3934/dcdsb.2018244

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, Y., Chen, F., Zhong, L.: Stability analysis of a prey–predator model with holling type III response function incorporating a prey refuge. Appl. Math. Comput. 182, 672–683 (2006). https://doi.org/10.1016/j.amc.2006.04.030

    Article  MathSciNet  MATH  Google Scholar 

  14. Lamontagne, Y., Coutu, C., Rousseau, C.: Bifurcation analysis of a predator–prey system with generalised Holling type III functional response. J. Dyn. Differ. Equ. 20, 535–571 (2008). https://doi.org/10.1007/s10884-008-9102-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Ding, W., Hendon, R., Cathey, B., Lancaster, E., Germick, R.: Discrete time optimal control applied to pest control problems. Involve J. Math. 7, 479–489 (2014). https://doi.org/10.2140/involve.2014.7.479

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, R.L., Wang, W.X., Qin, L.J.: Optimal harvesting policy of discrete-time predator–prey dynamic system with Holling type-IV functional response and its simulation. Appl. Comput. Math. 4, 20–29 (2015)

    Article  Google Scholar 

  17. Wu, T.: Dynamic behaviors of a discrete two species predator–prey system incorporating harvesting. Discrete Dyn. Nat. Soc. (2012). https://doi.org/10.1155/2012/429076

    Article  MathSciNet  MATH  Google Scholar 

  18. Hochard, J., Finnoff, D.: Gray wolf population projection with intraspecific competition. Nat. Resour. Model. 27, 360–375 (2014)

    Article  MathSciNet  Google Scholar 

  19. Jung, T.S., Czetwertynski, S.M.: Niche overlap and the potential for competition between reintroduced bison and other ungulates in southwestern Yukon, Yukon Fiah and Wildlife Branch report TR-13-15 (2013). https://www.sciencebase.gov/catalog/item/5771b8abe4b07657d1a6ccf2

  20. Hansen, R.M., Reid, L.D.: Diet overlap of deer, elk, and cattle in Southern Colorado. J. Range Manag. 28, 43–47 (1975)

    Article  Google Scholar 

  21. Jung, T.S., Stotyn, S.A., Czetwertynski, S.M.: Dietary overlap and potential competition in a dynamic ungulate community in Northwestern Canada. J. Wildl. Manag. 79, 1277–1285 (2015)

    Article  Google Scholar 

  22. Vila, A.R., Borrelli, L., Martinez, L.: Dietary overlap between huemul and livestock in Los Alerces National Park, Argentina. J. Wildl. Manag. 73, 368–373 (2010)

    Article  Google Scholar 

  23. Banerjee, R., Das, P., Mukherjee, D.: Stability and permanence of a discrete-time two-prey one-predator system with Holling type-III functional response. Chaos Solitons Fractals 117, 240–248 (2018)

    Article  MathSciNet  Google Scholar 

  24. Wang, L., Wang, M.: Ordinary Difference Equation. Xinjiang University Press, Xinjiang (1991)

    Google Scholar 

  25. Hwang, C.L., Gong, Y., Fan, L.T.: A discrete version of Pontryagin’s maximum principle. Oper. Res. 15, 139–146 (1967). https://doi.org/10.1287/opre.15.1.139

    Article  MathSciNet  MATH  Google Scholar 

  26. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models. Mathematical and Computational Biology, 1st edn. Chapman & Hall/CRC, London (2007)

    Book  Google Scholar 

  27. Banerjee, R., Das, P., Mukherjee, D.: Stability and permanent co-existence of a discrete-time two-prey one-predator system with harvesting effort. AIP Conf. Proc. 2159, 030002 (2019). https://doi.org/10.1063/1.5127467

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India

    Ritwick Banerjee & Pritha Das

  2. Department of Mathematics, Vivekananda College, Thakurpukur, Kolkata, 700063, India

    Debasis Mukherjee

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  1. Ritwick Banerjee
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  2. Pritha Das
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Correspondence to Ritwick Banerjee.

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Banerjee, R., Das, P. & Mukherjee, D. Global dynamics of a Holling Type-III two prey–one predator discrete model with optimal harvest strategy. Nonlinear Dyn 99, 3285–3300 (2020). https://doi.org/10.1007/s11071-020-05490-0

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